# Questions tagged [legendre-polynomials]

The legendre-polynomials tag has no usage guidance.

48
questions

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### Product of d-dimensional Legendre polynomials

Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...

0
votes

1
answer

132
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### How are the Legendre Polynomials of second kind for negative degrees defined?

For a script I have to evaluate the associated Legendre polynomial of second kind $Q^0_{n}(z)$.
Until now, I was using an implementation that is based on the definition in equation 8.702 in the Book &...

1
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0
answers

103
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### Algorithm for converting from 2D Legendre basis to 2D Monomial basis

I am dealing with function written in a 2D Legendre polynomial basis and I need to convert it so that it's written in a 2D monomial basis. I've found of an algorithm that allows for change of basis ...

3
votes

0
answers

245
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### Derivation of an integral containing the complete elliptic integral of the first kind

I found the following formula in "INTEGRALS AND SERIES, vol.3" by Prudnikov, Brychkov and Marichev (page 188, eq.5).
$$\int_0^{\infty} \frac{x^{\alpha-1}}{\sqrt{(a+x)^2+z^2}}K(\frac{2\sqrt{...

3
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0
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191
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### How to calculate the integral of a product of a spherical Hankel function with associated Legendre polynomials

From numerical experiments in Mathematica, I have found the following expression for the integral:
$$
\int_{-1}^{1}h_{n}^{(1)}\left(\sqrt{a^{2}+b^{2}+2ab\tau}\right)P_{n}^{m}\left(\frac{a\tau+b}{\sqrt{...

0
votes

0
answers

125
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### What's the convergence condition for the generating function formula of Legendre polynomials?

What is the convergence condition of the next infinite series about the Legendre polynomials $P_n(x)$?
$$
\frac{1}{\sqrt{1-2xt+t^2}}=\sum_{n=0}^\infty P_n(x)t^n
$$
I know it is convergent at least ...

3
votes

1
answer

442
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### Generating function of the product of Legendre polynomials

The generating function of the product of Legendre polynomials for the same $n$ is given by
\begin{aligned}
\sum_{n=0}^{\infty} z^{n} \mathrm{P}_{n}(\cos \alpha) \mathrm{P}_{n}(\cos \beta)&=\frac{\...

0
votes

1
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177
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### Properties of a function $C_\ell(\ell)$ which checks an inequality in ideal case (decreasing assumption) and after estimating impact in general case

Suppose that
$X=2 \ell+1, Y=C_{\ell}$, both $X$ and $Y$ are function of $\ell$, $X$ is increasing and $Y$ is assuming to be decreasing.
But in reality, my data follow a $C_\ell$ increasing for a small ...

0
votes

1
answer

156
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### Defining Legendre polynomials in terms of a sinusoidal function for $|x| \leq 1$

Would it be possible to define Legendre polynomials in terms of a sinusoidal function for $|x|\leq 1$ in a similar manner to Chebyshev polynomials being defined as $T_n(x) = \cos(n \cos^{-1}(x))$?
...

5
votes

2
answers

3k
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### Proof of spherical harmonic addition theorem

(Reposted from here and will be removed on this site if answered on MSE)
I have been trying to prove that
$$
P_\ell(\cos\gamma)=\frac{4\pi}{2\ell+1}\sum_{m=-\ell}^\ell Y_{\ell m}^{*}(\theta', \phi')Y_{...

4
votes

3
answers

1k
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### How to obtain the asymptotics of Legendre polynomials directly from their generating function

I'm reading about Legendre polynomials for additional information since it is interesting to know! Moreover it would help me with
a task I am working on. See
https://math.stackexchange.com/questions/...

2
votes

0
answers

130
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### Integral of Legendre's function

Is there any formula for computing the following integral $$\int_a^1(P^m_l)^2(x)\,dx \, ,\text{with} -1<a<1$$
where $P^m_l$ is the associated Legendre's function (of the first kind) of order $m\...

3
votes

1
answer

364
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### Closed form for the integral of a squared Legendre function

Is there a closed form for the integral $$\int_0^{\pi/2}(P_\nu^\mu(\cos\theta))^2\,\mathrm d\theta,\quad\mu>\nu\gt-\frac12$$ where $P_\nu^\mu(x)$ is the associated Legendre function of the first ...

2
votes

1
answer

379
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### Computing the integral $\int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)$

I would like to compute the following integral:
$$
I_\ell(\alpha) := \int_{-1}^1 dx \, |x| J_0(\alpha \sqrt{1 - x^2}) P_\ell(x)
\tag{1}
\label{1}
$$
where $\alpha \geq 0$, $J_0$ is the zeroth-order ...

2
votes

2
answers

410
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### A recurrence formula for the Legendre function $P_\mu^\nu(x)$

Im looking for a recurrence formula of type:
$$(\mu-\nu) x P_\mu^\nu(x) + P_{\mu-1}^\nu(x)=?, \quad \mu,\nu\in \mathbb R$$
where $P_\mu^\nu(x)$ is the Legendre function of the first kind (solution to ...

5
votes

0
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254
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### $L^p$-convergence of truncated Legendre series approximation to inverse error function $\text{erfinv}$

Bounding the $L^p$-error for an $n$-th order Legendre series approximation
I have function $f\,\colon (-1, 1) \to \mathbb{R}$ where $f \in L^q(-1, 1)$ for any $q \geq 1$. I approximate $f$ using a ...

0
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0
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177
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### Any good references on the decay rate of Legendre coefficient?

Let $P_n:[-1,1]\rightarrow\mathbb{R}$ be the $n$-th Legendre Polynomial. and, let
$$a_n:=\int_{-1}^1 f(t) P_n(t)\, dt$$
for some $f:[-1,1]\rightarrow\mathbb{R}$.
Are there any good references on the ...

5
votes

1
answer

546
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### Gegenbauer's addition theorem for Jacobi polynomials

I have the following identity,
$$\int_{-1}^{1} \! dz \, j_0\bigl(\sqrt{x^2 + y^2 - 2xy z}\bigr) \, P_n(z)
= 2 \, j_n(x) \, j_n(y) \;,$$
where $x, y > 0$, $P_n$ is a Legendre polynomial, and $...

1
vote

1
answer

416
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### Generate a two-variable polynomial from its "roots [closed]

I know, from mathematics basis, that a polynomial with one variable can be factor in function of its roots, so I can generate any one-variable polynomial from its zeros.
But I want know if is ...

5
votes

1
answer

2k
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### Convergence of the series of Legendre polynomials

Consider the generating function of Legendre polynomials:
$$\frac{1}{\sqrt{1 - 2xt + t^2}} = \sum\limits^{\infty}_{n=0} P_n(x)t^n$$
Is it true that for $0<x<1, t=1$ series of Legendre ...

0
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0
answers

83
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### Orthogonal functions and linear operators

Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}...

4
votes

3
answers

2k
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### Legendre Polynomial Integral over half space

I need to compute the following integral
$$
I_{n,m} := \int_0^1 P_n(x) P_m(x) \; \mathrm{d}x
$$
where $P_n$ is the Legendre polynomial.
For an even sum $n+m=2l$ it is easy to show that
$$
I_{n,m} = \...

4
votes

1
answer

203
views

### Integral with Legendre polynomial

Let $n$ be an integer $\geq 2$ and let $p_n(x)=\displaystyle \frac{1}{n!}(x^n(1-x)^n)^{(n)}$ be a Legendre polynomial.
Let $k $ be a positive integer.
I would like know if there is a closed formula (...

3
votes

1
answer

232
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### Looking for bound in integral involving Legendre polynomial

I'm looking for an upper bound to the following integral or equivalent when $n$ leads to $ +\infty $ to the following expression
$$I_n:=\left|\int_{0}^1 \int_{0}^1 \frac{p_n(x) p_n(y)}{(1-xy)} dx dy ...

1
vote

2
answers

778
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### Integral formula involving Legendre polynomial

I would like a proof of the following equation below, where $(P_n)$ denotes Legendre polynomials. I guess this formula to hold; I checked the first several values.
\begin{equation}
\int_{-1}^{1}\sqrt{...

3
votes

2
answers

430
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### Legendre equation: An interpretation [closed]

I am a student of physics and, especially in quantum mechanics, we are presented with the Legendre equation:
\begin{eqnarray}
(1-x^2)y''-2xy'+l(l+1)y=0.
\end{eqnarray}
Doing some calculations, we ...

2
votes

1
answer

207
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### Integration of hypergeometric product for legendre polynomials

I'm looking for a general solution to the integral:
$\int_{0}^1 {_2}F_1(m-n,m+n+1,m+1;z){_2}F_1(m-k+1,m+k+2,m+2;z) dz$
where $m,n,k\in \mathbb{N}$ and $m\leqslant n$ and $m+1 \leqslant k$.
To give ...

2
votes

0
answers

110
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### Rate of convergence of generalized polynomial chaos

Let $\eta=g(\xi_1,\ldots,\xi_M)$ be a random variable expressed as a function of the random vector $\xi=(\xi_1,\ldots,\xi_M)$. Assume that $\xi_1,\ldots,\xi_M$ are absolutely continuous and ...

3
votes

0
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### Identity for the product of two different associated Legendre polynomials

In the answer to Clausen’s identity for associated Legendre polynomials the following result was indicated:
$$
\small{\left(P_n^m(\cos\theta)\right)^2=(\sin{\theta})^{2m}\frac{(m+n)!}{(n-m)!}\sum_{k=0}...

2
votes

1
answer

625
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### Clausen’s identity for associated Legendre polynomials

Clausen’s identity for Legendre polynomials has the form (see, for example,
A generating function of the squares of Legendre polynomials, by Wadim Zudilin: https://arxiv.org/abs/1210.2493)
$$P_n(\cos{\...

3
votes

1
answer

706
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### Bounds on Legendre polynomials on the complex plane

Let $P_n$ be the Legendre polynomial of degree $n$. Could you suggest me some references to bound the polynomials on the complex plane (near the real line in particular) ? More specifically, I need to ...

5
votes

2
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3k
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### Relation between Legendre and Chebyshev polynomials

Where I could find relationships between Legendre and Chebyshev polynomials?
For example I found with maple
$$ P_n(\cos\theta)=\sum_{k=0}^n(-1)^{n+k}\frac{2-\delta_{k0}}{4^n}
\binom{n-k}{\frac{n-k}{2}...

3
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0
answers

469
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### Formula of Laplace for the asymptotic expansion of Legendre polynomials

According to the so-called formula of Laplace (see, e.g., [1, Theorem 8.21.2]), Legendre polynomials admit the following asymptotic expansion:
$$
P_n(\cos\theta) = \sqrt{2} (\pi n\sin\theta)^{-\frac12}...

1
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0
answers

498
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### Legendre functions as Hypergeometric functions

Is there some approximation or regularization that goes in tacitly in the following equality:
$Q_{\lambda }(z) = \frac{\sqrt{\pi } \Gamma (\lambda +1)}{2^{\lambda +1} \Gamma \left(\lambda +\frac{3}{2}...

2
votes

0
answers

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### Legendre expansion of $r(x) = f(x)/g(x)$ using a finite number of samples from $f(x)$ and $g(x)$

I have two finite sets of events $\{x_1, ..., x_N\}$ and $\{y_1, ..., y_N\}$ that are sampled from the PDFs $f(x)$ and $g(x)$, respectively, where $x \in [-1,+1]$. I want to estimate the Legendre ...

1
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0
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### Expansion of prolate spheroidal harmonics

For two coordinate frames $O'$ and $O''$ both offset along the $z$-axis by $\pm R$ respectively, with corresponding offset spherical coordinates $r'$, $\theta'$, $r''$ and $\theta''$, and with prolate ...

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2
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1k
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### Recurrence of Legendre polynomial roots/ quadrature points

Consider Legendre polynomials $p_n (x)$ on $[-1,1]$. For each $n \in \mathbb{N}$ we denote the zeros of $p_n (x)$ by $\left( x_j ^{(n)} \right) _{j=1} ^n$.
We know that these roots are distinct, and ...

0
votes

1
answer

1k
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### Integrals involving associated legendre polynomials

Do the following integrals have a closed-form solution for any integer value of $m,l,k$ and $n$?
$\int^{\pi}_{0} P^{m}_{l}\left(\cos\theta\right)P^{n}_{k}\left(\cos\theta\right)\cot\theta d\theta$
$\...

7
votes

2
answers

5k
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### Legendre Polynomial Integral

How can I evaluate
$$ \int_{-1}^1 P_n(x)P_l(x)x^k dx $$
when $k$ is even?
Or what might be a source where I could find integrals like this?

6
votes

2
answers

754
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### Symmetric matrix formula for Gauss-Legendre quadrature

While searching the web, I came across the following algorithm for the Gauss-Legendre quadrature. I wasn't able to find a reference or a proof of my own as to why it works. I'll present it, and the ...

4
votes

1
answer

784
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### Reference for the exponential decay of Legendre coefficients

In Short: I look for a reference to the proof that the spectral coefficients in the Legendre (or Jacobi) expansion are of exponential decay rate.
Longer: If $p_n$ is the $n$-th Legendre polynomial, ...

1
vote

0
answers

472
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### Legendre equation with homogeneous boundary condition

I'm looking for solutions to the Legendre differential equation
$$
\frac{d}{dx}\left((1-x^2)\frac{d}{dx}f(x)\right)+(\alpha-1)\alpha f(x)=0
$$
with boundary conditions $f'(0)=0$ and $f(1)=0$, but ...

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votes

4
answers

1k
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### Maximum of the Vandermonde determinant / minimum of the logarithmic energy

The problem is to find the asymptotics (as $n\to\infty$) of the maximum (say $M_n$) of the Vandermonde determinant
$$V_n:=\prod_{0\le i<j\le n-1}(a_j-a_i)
$$
over all $a_0,\dots,a_{n-1}$ such ...

14
votes

2
answers

1k
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### Computing Gauss Legendre quadrature for large $N$

I've been scanning across the web, and haven't found a good method to compute the Gauss Legendre abscissas and weights $\{ x_j, w^j \} _{j=1}^N$ for large $N\in\mathbb{N}$. My question is how to do it,...

0
votes

1
answer

134
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### Equality cannot hold unless $x \in \{-1,1\}$ and/or Wronskian is not zero [closed]

By playing around with assoc. Legendre polynomials, I arrived at
$$((l+1)+m) (P_l^m(x))^2+((l+1)-m)(P_{l+1}^m(x))^2 = 2(l+1)x P_l^m(x)P_{l+1}^m(x).$$
Now, I want to show that we don't have equality ...

0
votes

1
answer

472
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### Integral Transform with associated Legendre Function of second kind as kernel

In my research the following equation appeared:
$$\frac{1}{4\pi}\int_{0}^{1}\frac{t^{s-1}(1-t)^{s-1}}{(\rho-t)^s}dt=\int_0^{\infty} f(a) Q^{i\sqrt{a}}_{s-1}(2\rho-1) da,$$
where $\rho,s>1$, $Q^{\...

6
votes

0
answers

287
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### Legendre polynomials and formal groups

Let $P_n(x)$ be Legendre polynomials:
$$\frac{1}{\sqrt{1-2tx+t^2}}=\sum\limits_{n=0}^{\infty}P_n(x)t^n.$$
Usual arguments from the theory of formal groups allow to
prove that for any $n$
$$P_n(x)=Q_n(...

1
vote

0
answers

137
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### Fourier-Legendre coefficients and Sobolev regularity

Let $f$ be an $L^1$-function supported in $[-b, b]$, where $0 < b \le b_0 < 1$. Let $Q_n$ be the normalised Legendre polynomials and let
$a_n = \int f(x) Q_n(x) dx$ be the Fourier-Legendre ...